American Institute of Mathematical Sciences

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doi: 10.3934/dcdsb.2021065

Single-target networks

Gheorghe Craciun 1, , Jiaxin Jin 2,, and  Polly Y. Yu 2,

1. 

Department of Mathematics and Department of Biomolecular Chemistry, University of Wisconsin-Madison, 53706
2. 

Department of Mathematics, University of Wisconsin-Madison, 53706

* Corresponding author: Jiaxin Jin

Received  June 2020 Revised  December 2020 Published  March 2021

Fund Project: The authors are supported by NSF grant DMS-1816238

Reaction networks can be regarded as finite oriented graphs embedded in Euclidean space. Single-target networks are reaction networks with an arbitrarily set of source vertices, but only one sink vertex. We completely characterize the dynamics of all mass-action systems generated by single-target networks, as follows: either (i) the system is globally stable for all choice of rate constants (in fact, is dynamically equivalent to a detailed-balanced system with a single linkage class), or (ii) the system has no positive steady states for any choice of rate constants and all trajectories must converge to the boundary of the positive orthant or to infinity. Moreover, we show that global stability occurs if and only if the target vertex of the network is in the relative interior of the convex hull of the source vertices.

Keywords: Dynamical systems in biology, kinetics in biochemical problems, systems biology, networks, chemical kinetics in thermodynamics and heat transfer.
Mathematics Subject Classification: Primary: 37N25, 92C42; Secondary: 80A30.
Citation: Gheorghe Craciun, Jiaxin Jin, Polly Y. Yu. Single-target networks. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021065
References:
[1]

D. F. Anderson, A proof of the global attractor conjecture in the single linkage class case, SIAM Journal on Applied Mathematics, 71 (2011), 1487-1508.  doi: 10.1137/11082631X.  Google Scholar

[2]

D. F. Anderson, J. D. Brunner, G. Craciun and M. D. Johnston, On classes of reaction networks and their associated polynomial dynamical systems, (2020). Google Scholar

[3]

D. Angeli, A tutorial on chemical reaction network dynamics, European Journal of Control, 15 (2009), 398-406.  doi: 10.3166/ejc.15.398-406.  Google Scholar

[4]

M. W. Birch, Maximum likelihood in three-way contingency tables, Journal of the Royal Statistical Society. Series B (Methodological), 25 (1963), 220-233.  doi: 10.1111/j.2517-6161.1963.tb00504.x.  Google Scholar

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B. Boros, Existence of positive steady states for weakly reversible mass-action systems, SIAM Journal on Mathematical Analysis, 51 (2019), 435-449.  doi: 10.1137/17M115534X.  Google Scholar

[6]

B. Boros and J. Hofbauer, Permanence of weakly reversible mass-action systems with a single linkage class, SIAM Journal on Applied Dynamical Systems, 19 (2020), 352-365.  doi: 10.1137/19M1248431.  Google Scholar

[7]

M. L. Brustenga, G. Craciun and M-S Sorea, Disguised toric dynamical systems, (2020). Google Scholar

[8]

G. Craciun, Toric differential inclusions and a proof of the global attractor conjecture, (2015), arXiv: 1501.02860 [math.DS]. Google Scholar

[9]

G. Craciun, Polynomial dynamical systems, reaction networks, and toric differential inclusions, SIAM Journal on Applied Algebra and Geometry, 3 (2019), 87-106.  doi: 10.1137/17M1129076.  Google Scholar

[10]

G. Craciun, A. Dickenstein, B. Sturmfels and A. Shiu, Toric dynamical systems, Journal of Symbolic Computation, 44 (2009), 1551-1565.  doi: 10.1016/j.jsc.2008.08.006.  Google Scholar

[11]

G. Craciun, J. Jin and P. Y. Yu, An efficient characterization of complex-balanced, detailed-balanced, and weakly reversible systems, SIAM Journal on Applied Mathematics, 80 (2020), 183-205.  doi: 10.1137/19M1244494.  Google Scholar

[12]

G. Craciun, J. Jin and P. Y. Yu, Dynamical equivalence to complex balancing as an open condition in parameter space, in Preparation. Google Scholar

[13]

G. Craciun, F. Nazarov and C. Pantea, Persistence and permanence of mass-action and power-law dynamical systems, SIAM Journal on Applied Mathematics, 73 (2013), 305-329.  doi: 10.1137/100812355.  Google Scholar

[14]

G. Craciun and C. Pantea, Identifiability of chemical reaction networks, Journal of Mathematical Chemistry, 44 (2008), 244-259.  doi: 10.1007/s10910-007-9307-x.  Google Scholar

[15]

A. Dickenstein and M. Pérez Millán, How far is complex balancing from detailed balancing?, Bulletin of Mathematical Biology, 73 (2011), 811-828.  doi: 10.1007/s11538-010-9611-7.  Google Scholar

[16]

M. Feinberg, Complex balancing in general kinetic systems, Archive for Rational Mechanics and Analysis, 49 (1972), 187-194.  doi: 10.1007/BF00255665.  Google Scholar

[17]

M. Feinberg, Chemical reaction network structure and the stability of complex isothermal reactors - I. The Deficiency Zero and the Deficiency One Theorems, Chemical Engineering Science, 42 (1987), 2229-2268.   Google Scholar

[18]

M. Feinberg, Necessary and sufficient conditions for detailed balancing in mass action systems of arbitrary complexity, Chemical Engineering Science, 44 (1989), 1819-1827.   Google Scholar

[19]

M. Feinberg, Foundations of Chemical Reaction Network Theory, Applied Mathematical Sciences, Springer International Publishing, 2019.  Google Scholar

[20]

M. Gopalkrishnan, E. Miller and A. Shiu, A geometric approach to the global attractor conjecture, SIAM Journal on Applied Dynamical Systems, 13 (2014), 758-797.  doi: 10.1137/130928170.  Google Scholar

[21]

J. Gunawardena, Chemical reaction network theory for in-silico biologists, (2003), http://vcp.med.harvard.edu/papers/crnt.pdf, Google Scholar

[22]

F. Horn, Necessary and sufficient conditions for complex balancing in chemical kinetics, Archive for Rational Mechanics and Analysis, 49 (1972), 172-186.  doi: 10.1007/BF00255664.  Google Scholar

[23]

F. Horn and R. Jackson, General mass action kinetics, Archive for Rational Mechanics and Analysis, 47 (1972), 81-116.  doi: 10.1007/BF00251225.  Google Scholar

[24]

M. D. Johnston, Translated chemical reaction networks, Bulletin of Mathematical Biology, 76 (2014), 1081-1116.  doi: 10.1007/s11538-014-9947-5.  Google Scholar

[25]

M. D. Johnston and E. Burton, Computing weakly reversible deficiency zero network translations using elementary flux modes, Bulletin of Mathematical Biology, 81 (2019), 1613-1644.  doi: 10.1007/s11538-019-00579-z.  Google Scholar

[26]

M. D. Johnston, D. Siegel and G. Szederkényi, Computing weakly reversible linearly conjugate chemical reaction networks with minimal deficiencys, Mathematical Biosciences, 241 (2013), 88-98.  doi: 10.1016/j.mbs.2012.09.008.  Google Scholar

[27]

G. Lipták, G. Szederkényi and K. M. Hangos, Computing zero deficiency realizations of kinetic systems, Systems & Control Letters, 81 (2015), 24-30.  doi: 10.1016/j.sysconle.2015.05.001.  Google Scholar

[28]

S. Müller and G. Regensburger, Generalized mass action systems: Complex balancing equilibria and sign vectors of the stoichiometric and kinetic-order subspaces, SIAM Journal on Applied Mathematics, 72 (2012), 1926-1947.  doi: 10.1137/110847056.  Google Scholar

[29]

L. Onsager, Reciprocal relations in irreversible processes I., Physical Review, 37 (1931), 405-426.   Google Scholar

[30] L. Pachter and B. Sturmfels, Algebraic Statistics for Computational Biology, Cambridge University Press, 2005.  doi: 10.1017/CBO9780511610684.007.  Google Scholar
[31]

C. Pantea, On the persistence and global stability of mass-action systems, SIAM Journal on Mathematical Analysis, 44 (2012), 1636-1673.  doi: 10.1137/110840509.  Google Scholar

[32]

J. Rudan, G. Szederkényi, K. M. Hangos and T. Péni, Polynomial time algorithms to determine weakly reversible realizations of chemical reaction networks, Journal of Mathematical Chemistry, 52 (2014), 1386-1404.  doi: 10.1007/s10910-014-0318-0.  Google Scholar

[33]

S. Schuster and R. Schuster, A generalization of Wegscheider's condition. Implications for properties of steady states and for quasi-steady-state approximation, Journal of Mathematical Chemistry, 3 (1989), 25-42.  doi: 10.1007/BF01171883.  Google Scholar

[34]

G. Szederkényi, Comment on "Identifiability of chemical reaction networks" by G. Craciun and C. Pantea, Journal of Mathematical Chemistry, 45 (2009), 1172-1174.  doi: 10.1007/s10910-008-9499-8.  Google Scholar

[35]

G. Szederkényi, J. R. Banga and A. A. Alonso, CRNreals: A toolbox for distinguishability and identifiability analysis of biochemical reaction networks, Bioinformatics, 28 (2012), 1549-1550.   Google Scholar

[36]

G. Szederkényi and K. M. Hangos, Finding complex balanced and detailed balanced realizations of chemical reaction networks, Journal of Mathematical Chemistry, 49 (2011), 1163-1179.  doi: 10.1007/s10910-011-9804-9.  Google Scholar

[37]

A. I. Vol'pert, Differential equations on graphs, Math. USSR-Sb, 88 (1972), 578-588.   Google Scholar

[38]

R. Wegscheider, Über simultane gleichgewichte und die beziehungen zwischen thermodynamik und reactionskinetik homogener systeme, Monatshefte für Chemie und verwandte Teile anderer Wissenschaften, 22 (1901), 849-906.   Google Scholar

[39]

P. Y. Yu and G. Craciun, Mathematical analysis of chemical reaction systems, Israel Journal of Chemistry, 58 (2018), 733-741.   Google Scholar

show all references

References:
[1]

D. F. Anderson, A proof of the global attractor conjecture in the single linkage class case, SIAM Journal on Applied Mathematics, 71 (2011), 1487-1508.  doi: 10.1137/11082631X.  Google Scholar

[2]

D. F. Anderson, J. D. Brunner, G. Craciun and M. D. Johnston, On classes of reaction networks and their associated polynomial dynamical systems, (2020). Google Scholar

[3]

D. Angeli, A tutorial on chemical reaction network dynamics, European Journal of Control, 15 (2009), 398-406.  doi: 10.3166/ejc.15.398-406.  Google Scholar

[4]

M. W. Birch, Maximum likelihood in three-way contingency tables, Journal of the Royal Statistical Society. Series B (Methodological), 25 (1963), 220-233.  doi: 10.1111/j.2517-6161.1963.tb00504.x.  Google Scholar

[5]

B. Boros, Existence of positive steady states for weakly reversible mass-action systems, SIAM Journal on Mathematical Analysis, 51 (2019), 435-449.  doi: 10.1137/17M115534X.  Google Scholar

[6]

B. Boros and J. Hofbauer, Permanence of weakly reversible mass-action systems with a single linkage class, SIAM Journal on Applied Dynamical Systems, 19 (2020), 352-365.  doi: 10.1137/19M1248431.  Google Scholar

[7]

M. L. Brustenga, G. Craciun and M-S Sorea, Disguised toric dynamical systems, (2020). Google Scholar

[8]

G. Craciun, Toric differential inclusions and a proof of the global attractor conjecture, (2015), arXiv: 1501.02860 [math.DS]. Google Scholar

[9]

G. Craciun, Polynomial dynamical systems, reaction networks, and toric differential inclusions, SIAM Journal on Applied Algebra and Geometry, 3 (2019), 87-106.  doi: 10.1137/17M1129076.  Google Scholar

[10]

G. Craciun, A. Dickenstein, B. Sturmfels and A. Shiu, Toric dynamical systems, Journal of Symbolic Computation, 44 (2009), 1551-1565.  doi: 10.1016/j.jsc.2008.08.006.  Google Scholar

[11]

G. Craciun, J. Jin and P. Y. Yu, An efficient characterization of complex-balanced, detailed-balanced, and weakly reversible systems, SIAM Journal on Applied Mathematics, 80 (2020), 183-205.  doi: 10.1137/19M1244494.  Google Scholar

[12]

G. Craciun, J. Jin and P. Y. Yu, Dynamical equivalence to complex balancing as an open condition in parameter space, in Preparation. Google Scholar

[13]

G. Craciun, F. Nazarov and C. Pantea, Persistence and permanence of mass-action and power-law dynamical systems, SIAM Journal on Applied Mathematics, 73 (2013), 305-329.  doi: 10.1137/100812355.  Google Scholar

[14]

G. Craciun and C. Pantea, Identifiability of chemical reaction networks, Journal of Mathematical Chemistry, 44 (2008), 244-259.  doi: 10.1007/s10910-007-9307-x.  Google Scholar

[15]

A. Dickenstein and M. Pérez Millán, How far is complex balancing from detailed balancing?, Bulletin of Mathematical Biology, 73 (2011), 811-828.  doi: 10.1007/s11538-010-9611-7.  Google Scholar

[16]

M. Feinberg, Complex balancing in general kinetic systems, Archive for Rational Mechanics and Analysis, 49 (1972), 187-194.  doi: 10.1007/BF00255665.  Google Scholar

[17]

M. Feinberg, Chemical reaction network structure and the stability of complex isothermal reactors - I. The Deficiency Zero and the Deficiency One Theorems, Chemical Engineering Science, 42 (1987), 2229-2268.   Google Scholar

[18]

M. Feinberg, Necessary and sufficient conditions for detailed balancing in mass action systems of arbitrary complexity, Chemical Engineering Science, 44 (1989), 1819-1827.   Google Scholar

[19]

M. Feinberg, Foundations of Chemical Reaction Network Theory, Applied Mathematical Sciences, Springer International Publishing, 2019.  Google Scholar

[20]

M. Gopalkrishnan, E. Miller and A. Shiu, A geometric approach to the global attractor conjecture, SIAM Journal on Applied Dynamical Systems, 13 (2014), 758-797.  doi: 10.1137/130928170.  Google Scholar

[21]

J. Gunawardena, Chemical reaction network theory for in-silico biologists, (2003), http://vcp.med.harvard.edu/papers/crnt.pdf, Google Scholar

[22]

F. Horn, Necessary and sufficient conditions for complex balancing in chemical kinetics, Archive for Rational Mechanics and Analysis, 49 (1972), 172-186.  doi: 10.1007/BF00255664.  Google Scholar

[23]

F. Horn and R. Jackson, General mass action kinetics, Archive for Rational Mechanics and Analysis, 47 (1972), 81-116.  doi: 10.1007/BF00251225.  Google Scholar

[24]

M. D. Johnston, Translated chemical reaction networks, Bulletin of Mathematical Biology, 76 (2014), 1081-1116.  doi: 10.1007/s11538-014-9947-5.  Google Scholar

[25]

M. D. Johnston and E. Burton, Computing weakly reversible deficiency zero network translations using elementary flux modes, Bulletin of Mathematical Biology, 81 (2019), 1613-1644.  doi: 10.1007/s11538-019-00579-z.  Google Scholar

[26]

M. D. Johnston, D. Siegel and G. Szederkényi, Computing weakly reversible linearly conjugate chemical reaction networks with minimal deficiencys, Mathematical Biosciences, 241 (2013), 88-98.  doi: 10.1016/j.mbs.2012.09.008.  Google Scholar

[27]

G. Lipták, G. Szederkényi and K. M. Hangos, Computing zero deficiency realizations of kinetic systems, Systems & Control Letters, 81 (2015), 24-30.  doi: 10.1016/j.sysconle.2015.05.001.  Google Scholar

[28]

S. Müller and G. Regensburger, Generalized mass action systems: Complex balancing equilibria and sign vectors of the stoichiometric and kinetic-order subspaces, SIAM Journal on Applied Mathematics, 72 (2012), 1926-1947.  doi: 10.1137/110847056.  Google Scholar

[29]

L. Onsager, Reciprocal relations in irreversible processes I., Physical Review, 37 (1931), 405-426.   Google Scholar

[30] L. Pachter and B. Sturmfels, Algebraic Statistics for Computational Biology, Cambridge University Press, 2005.  doi: 10.1017/CBO9780511610684.007.  Google Scholar
[31]

C. Pantea, On the persistence and global stability of mass-action systems, SIAM Journal on Mathematical Analysis, 44 (2012), 1636-1673.  doi: 10.1137/110840509.  Google Scholar

[32]

J. Rudan, G. Szederkényi, K. M. Hangos and T. Péni, Polynomial time algorithms to determine weakly reversible realizations of chemical reaction networks, Journal of Mathematical Chemistry, 52 (2014), 1386-1404.  doi: 10.1007/s10910-014-0318-0.  Google Scholar

[33]

S. Schuster and R. Schuster, A generalization of Wegscheider's condition. Implications for properties of steady states and for quasi-steady-state approximation, Journal of Mathematical Chemistry, 3 (1989), 25-42.  doi: 10.1007/BF01171883.  Google Scholar

[34]

G. Szederkényi, Comment on "Identifiability of chemical reaction networks" by G. Craciun and C. Pantea, Journal of Mathematical Chemistry, 45 (2009), 1172-1174.  doi: 10.1007/s10910-008-9499-8.  Google Scholar

[35]

G. Szederkényi, J. R. Banga and A. A. Alonso, CRNreals: A toolbox for distinguishability and identifiability analysis of biochemical reaction networks, Bioinformatics, 28 (2012), 1549-1550.   Google Scholar

[36]

G. Szederkényi and K. M. Hangos, Finding complex balanced and detailed balanced realizations of chemical reaction networks, Journal of Mathematical Chemistry, 49 (2011), 1163-1179.  doi: 10.1007/s10910-011-9804-9.  Google Scholar

[37]

A. I. Vol'pert, Differential equations on graphs, Math. USSR-Sb, 88 (1972), 578-588.   Google Scholar

[38]

R. Wegscheider, Über simultane gleichgewichte und die beziehungen zwischen thermodynamik und reactionskinetik homogener systeme, Monatshefte für Chemie und verwandte Teile anderer Wissenschaften, 22 (1901), 849-906.   Google Scholar

[39]

P. Y. Yu and G. Craciun, Mathematical analysis of chemical reaction systems, Israel Journal of Chemistry, 58 (2018), 733-741.   Google Scholar

Figure 1.  (a) A single-target network that is globally stable under mass-action kinetics. (b)–(c) Single-target networks with no positive steady states. (d) Not a single-target network
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Figure 2.  Consider subnetworks of (a) under mass-action kinetics, whose associated dynamics is given by (9). If the coefficient of $ {\boldsymbol{x}}^{ {\boldsymbol{y}}_1} $ in $ \dot{x} $ is positive and the coefficient of $ {\boldsymbol{x}}^{ {\boldsymbol{y}}_3} $ in $ \dot{x} $ is negative, then the system (9) can be realized by a single-target network, determined by the sign of $ {\boldsymbol{x}}^{ {\boldsymbol{y}}_2} $ in $ \dot{x} $. If the net directions are as shown in (b), then (9) can be realized by the single-target network in (c). Similarly, if the net directions appear as in (d), then (9) can be realized by the network in (e)
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Figure 3.  Reversible systems in (a) Example 3.12 and (b) Example 3.13 that are dynamically equivalent to detailed-balanced systems. Each undirected edge represents a pair of reversible edges
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Figure 4.  Geometric argument for dynamically equivalence to single-target network in Example 3.12. Shown are the edges with $ \mathrm{{X}}_1 + \mathrm{{X}}_2 $ as their source. The centre $ \left(\frac{1}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2} \right)^\top $ of the tetrahedron is marked in blue. With rate constants given in the example, the weighted sum of reaction vectors points from the source to the centre
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Figure 5.  (a) The mass-action system with two target vertices from Example 4.2, which is dynamically equivalent to a complex-balanced system using a subnetwork of (b) if and only if $ \frac{1}{25} \leq \frac{ \kappa_1 \kappa_3}{ \kappa_2 \kappa_4} \leq 25 $
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Figure 5(a) is dynamically equivalent to a complex-balanced system if and only if $ \frac{1}{25} \leq \frac{ \kappa_1 \kappa_3}{ \kappa_2 \kappa_4} \leq 25 $. The system is equivalent to (a) when $ \kappa_2 \kappa_4 = 25 \kappa_1 \kappa_3 $ and (b) when $ 25 \kappa_2 \kappa_4 = \kappa_1 \kappa_3 $. For $ \frac{1}{25} \leq \frac{ \kappa_1 \kappa_3}{ \kappa_2 \kappa_4} \leq 25 $, the dynamically equivalent system is an appropriate convex combination of (a) and (b)">Figure 6.  The system in Figure 5(a) is dynamically equivalent to a complex-balanced system if and only if $ \frac{1}{25} \leq \frac{ \kappa_1 \kappa_3}{ \kappa_2 \kappa_4} \leq 25 $. The system is equivalent to (a) when $ \kappa_2 \kappa_4 = 25 \kappa_1 \kappa_3 $ and (b) when $ 25 \kappa_2 \kappa_4 = \kappa_1 \kappa_3 $. For $ \frac{1}{25} \leq \frac{ \kappa_1 \kappa_3}{ \kappa_2 \kappa_4} \leq 25 $, the dynamically equivalent system is an appropriate convex combination of (a) and (b)
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